Tiegang Liu

Ghost fluid method for strong shock impacting on material interface

It is found that the original ghost fluid method (GFM) as put forth by Fedkiw et al. [J. Comp. Phys. 152 (1999) 457] does not work consistently and efficiently using isentropic fix when applied to a strong shock impacting on a material interface. In this work, the causes for such inapplicability of the original GFM are analysed and a modified GFM is proposed and developed for greater robustness and consistency. Numerical tests also show that the modified GFM has the property of reduced conservation error and is less problem-related.

An adaptive ghost fluid finite volume method for compressible gas-water simulations

An adaptive ghost fluid finite volume method is developed for one- and two-dimensional compressible multi-medium flows in this work. It couples the real ghost fluid method (GFM) [C.W. Wang, T.G. Liu, B.C. Khoo, A real-ghost fluid method for the simulation of multi-medium compressible flow, SIAM J. Sci. Comput. 28 (2006) 278-302] and the adaptive moving mesh method [H.Z. Tang, T. Tang. Moving mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487-515; H.Z. Tang, T. Tang, P.W. Zhang, An adaptive mesh redistribution method for non-linear Hamilton-Jacobi equations in two- and three-dimensions, J. Comput. Phys. 188 (2003) 543-572], and thus combines their advantages. This work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the GFM around the material interface and other discontinuities. Besides the improvement of flow field resolution, the adaptive GFM also largely increases the computational efficiency. Several numerical experiments are conducted to demonstrate robustness and efficiency of the current method. They include several 1D and 2D gas-water flow problems, involving a large density gradient at the material interface and strong shock-interface interactions. The results show that our algorithm can capture the shock waves and the material interface accurately, and is stable and robust even for solutions with large density and pressure gradients.

A note on the conservative schemes for the Euler equations

This note gives a numerical investigation for the popular high resolution conservative schemes when applied to inviscid, compressible, perfect gas flows with an initial high density ratio as well as a high pressure ratio. The results show that they work very inefficiently and may give inaccurate numerical results even over a very fine mesh when applied to such a problem. Numerical tests show that increasing the order of accuracy of the numerical schemes does not help much in improving the numerical results. How to cure this difficulty is still open.

Runge-Kutta discontinuous Galerkin methods for compressible two-medium flow simulations: One-dimensional case

The Runge-Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws is a high order finite element method, which utilizes the useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, TVD Runge-Kutta time discretizations, and limiters. In this paper, we investigate using the RKDG finite element method for compressible two-medium flow simulation with conservative treatment of the moving material interfaces. Numerical results for both gas-gas and gas-water flows in one-dimension are provided to demonstrate the characteristic behavior of this approach.

The Modified Ghost Fluid Method for Coupling of Fluid and Structure Constituted with Hydro-Elasto-Plastic Equation of State

In this work, the modified ghost fluid method (MGFM) [T. G. Liu, B. C. Khoo, and K. S. Yeo, J. Comput. Phys., 190 (2003), pp. 651-681] is further developed and applied to treat the compressible fluid-compressible structure coupling. To facilitate theoretical analysis, the structure is modeled as elastic-plastic material with perfect plasticity and constituted with the hydro-elasto-plastic equation of state [H. S. Tang and F. Sotiropoulos, J. Comput. Phys., 151 (1999), pp. 790-815] under strong impact. This results in the coupled compressible fluid-compressible structure system which is fully hyperbolic. To understand the effect of structure deformation on the interfacial and flow status, the compressible fluid-compressible structure Riemann problem is analyzed in the consideration of material deformation with an approximate Riemann problem solver proposed to take into account the effect of material elastic-plastic deformation. We clearly show the ghost fluid method can be applied to treat the flow-deformable structure coupling under strong impact provided that a proper Riemann problem solver is used to predict the ghost fluid states. And the resultant MGFM can work effectively and efficiently in such situations. Various examples are presented to validate and support the conclusions reached.

Implementation of the GRP scheme for computing radially symmetric compressible fluid flows

The study of radially symmetric compressible fluid flows is interesting both from the theoretical and numerical points of view. Spherical explosion and implosion in air, water and other media are well-known problems in application. Typical difficulties lie in the treatment of singularity in the geometrical source and the imposition of boundary conditions at the symmetric center, in addition to the resolution of classical discontinuities (shocks and contact discontinuities). In the present paper we present the implementation of direct generalized Riemann problem (GRP) scheme to resolve this issue. The scheme is obtained directly by the time integration of the fluid flows. Our new contribution is to show rigorously that the singularity is removable and derive the updating formulae for mass and energy at the center. Together with the vanishing of the momentum, we obtain new numerical boundary conditions at the center, which are then incorporated into the GRP scheme. The main ingredient is the passage from the Cartesian coordinates to the radially symmetric coordinates.

A direct discontinuous Galerkin method for the compressible NavierStokes equations on arbitrary grids

A Direct Discontinuous Galerkin (DDG) method is developed for solving the compressible NavierStokes equations on arbitrary grids in the framework of DG methods. The DDG method, originally introduced for scalar diffusion problems on structured grids, is extended to discretize viscous and heat fluxes in the NavierStokes equations. Two approaches of implementing the DDG method to compute numerical diffusive fluxes for the NavierStokes equations are presented: one is based on the conservative variables, and the other is based on the primitive variables. The importance of the characteristic cell size used in the DDG formulation on unstructured grids is examined. The numerical fluxes on the boundary by the DDG method are discussed. A number of test cases are presented to assess the performance of the DDG method for solving the compressible NavierStokes equations. Based on our numerical results, we observe that DDG method can achieve the designed order of accuracy and is able to deliver the same accuracy as the widely used BR2 method at a significantly reduced cost, clearly demonstrating that the DDG method provides an attractive alternative for solving the compressible NavierStokes equations on arbitrary grids owning to its simplicity in implementation and its efficiency in computational cost.

Analysis and development of adjoint-based h-adaptive direct discontinuous Galerkin method for the compressible Navier–Stokes equations

Abstract In this paper, an adjoint-based high-order h-adaptive direct discontinuous Galerkin method is developed and analyzed for the two dimensional steady state compressible Navier–Stokes equations. Particular emphasis is devoted to the analysis of the adjoint consistency for three different direct discontinuous Galerkin discretizations: including the original direct discontinuous Galerkin method (DDG), the direct discontinuous Galerkin method with interface correction (DDG(IC)) and the symmetric direct discontinuous Galerkin method (SDDG). Theoretical analysis shows the extra interface correction term adopted in the DDG(IC) method and the SDDG method plays a key role in preserving the adjoint consistency. To be specific, for the model problem considered in this work, we prove that the original DDG method is not adjoint consistent, while the DDG(IC) method and the SDDG method can be adjoint consistent with appropriate treatment of boundary conditions and correct modifications towards the underlying output functionals. The performance of those three DDG methods is carefully investigated and evaluated through typical test cases. Based on the theoretical analysis, an adjoint-based h-adaptive DDG(IC) method is further developed and evaluated, numerical experiment shows its potential in the applications of adjoint-based adaptation for simulating compressible flows.

A high order compact least-squares reconstructed discontinuous Galerkin method for the steady-state compressible flows on hybrid grids

Abstract In this paper, a class of new high order reconstructed DG (rDG) methods based on the compact least-squares (CLS) reconstruction [23] , [24] is developed for simulating the two dimensional steady-state compressible flows on hybrid grids. The proposed method combines the advantages of the DG discretization with the flexibility of the compact least-squares reconstruction, which exhibits its superior potential in enhancing the level of accuracy and reducing the computational cost compared to the underlying DG methods with respect to the same number of degrees of freedom. To be specific, a third-order compact least-squares rDG( p 1 p 2 ) method and a fourth-order compact least-squares rDG( p 2 p 3 ) method are developed and investigated in this work. In this compact least-squares rDG method, the low order degrees of freedom are evolved through the underlying DG( p 1 ) method and DG( p 2 ) method, respectively, while the high order degrees of freedom are reconstructed through the compact least-squares reconstruction, in which the constitutive relations are built by requiring the reconstructed polynomial and its spatial derivatives on the target cell to conserve the cell averages and the corresponding spatial derivatives on the face-neighboring cells. The large sparse linear system resulted by the compact least-squares reconstruction can be solved relatively efficient when it is coupled with the temporal discretization in the steady-state simulations. A number of test cases are presented to assess the performance of the high order compact least-squares rDG methods, which demonstrates their potential to be an alternative approach for the high order numerical simulations of steady-state compressible flows.